Sum of the first 10 terms of a GP is equal to the sum of the first 12 terms in the same GP. Sum of the first 15 terms is 31, what is the third term in the GP?

Sum of the first 10 terms of a GP is equal to the sum of the first 12 terms in the same GP. Sum of the first 15 terms is 31, what is the third term in the GP? Correct Answer 31

Sum of first 10 terms is equal to sum of first 12 terms.

Sum of first 12 terms = Sum of first 10 terms + 11^th term + 12^th term

11^th term + 12^th term = 0

Let 11^th term = k,

common ratio = r

12^th term will be = kr

k + kr = 0

k (1 + r) = 0

r = -1 as k cannot be zero

Common ratio = -1

Now, sum of 15 terms = a(r¹⁵ - 1)/(r - 1) = 31

⇒ a(-1¹⁵ -1)/(-1 -1) = 31

⇒ a = 31

∴ Third term = ar² = 31 × (-1)² = 31